Upside-Down Rayleigh–Marchenko (UD-RM)

• UD-RM is a reciprocal redatuming technique that swaps source and receiver roles to perform spatial integrations over the well-sampled source carpet.
• It reconstructs full-wavefield subsurface images by solving for focusing functions that inherently account for both internal and free-surface multiples.
• Practical validations demonstrate UD-RM's reliability in sparse seabed data settings, with accelerated performance achieved through self-supervised learning methods.

Searching arXiv for Upside-Down Rayleigh-Marchenko and closely related Marchenko/Rayleigh papers. Use arXiv search with query: "Upside-Down Rayleigh-Marchenko" Upside-Down Rayleigh–Marchenko (UD-RM) is a reciprocal version of the Rayleigh-Marchenko method in which all spatial integrals are performed over the source carpet rather than over the receiver boundary, with the explicit aim of making single-sided redatuming practical for seabed seismic acquisitions with irregular or sparse receiver geometries. In its practical acoustic formulation, UD-RM is a theoretically exact redatuming scheme under lossless, pressure-release free-surface conditions, provided that the source carpet is sufficiently sampled and that the required up/down-separated boundary wavefields are available; in later work, the estimation of the focusing functions was accelerated through self-supervised learning, while related Marchenko and inverse-scattering results supplied broader full-field and spectral interpretations of the method (Wang et al., 2024, Wang et al., 29 Jul 2025, Wapenaar et al., 2021).

1. Definition and methodological position

UD-RM emerged from the observation that conventional Rayleigh-Marchenko (RM) remains tied to spatial integrals over the receiver boundary, which in ocean-bottom settings is often the poorly sampled side of the acquisition. The central reparameterization in UD-RM is therefore reciprocal: source and receiver roles are swapped so that the boundary integrations are evaluated on the source carpet $\Gamma_S$, which is typically better sampled than the seabed receiver plane $\Gamma_R$. In the seabed setting described in the literature, sources are near the free surface, receivers are at the seabed, and focal points lie in the subsurface volume between or below these levels; “upside-down” refers precisely to this acquisition geometry and to the associated reformulation of the redatuming operators (Wang et al., 2024, Wang et al., 29 Jul 2025).

Within that framework, UD-RM reconstructs redatumed subsurface wavefields by solving for focusing functions that enforce causal focusing at the chosen focal points. This places it within the Marchenko family of single-sided methods, but with two distinguishing properties. First, it inherits the Rayleigh-Marchenko treatment of separate source and receiver levels and of free-surface multiples. Second, it avoids explicit spatial integration over the sparse receiver side. The method can be interpreted as a full-wavefield extension of the mirror imaging method commonly used in seabed settings, but it is not limited to single-scattering mirror assumptions; it solves for focusing functions and reconstructs wavefields that include both internal and surface-related multiple interactions (Wang et al., 2024).

A recurrent misconception is that UD-RM is merely a practical remapping of mirror imaging. The cited work does not support that reduction. The method is presented instead as a full-wavefield redatuming scheme whose outputs are amplitude-friendly and whose multiple handling is intrinsic to the focusing-function formulation. Another misconception is that “exact” means unconditional. In the practical acoustic setting, exactness is explicitly tied to stated boundary conditions, sampling, separation, and windowing assumptions rather than claimed as a universal property (Wang et al., 2024).

2. Acquisition geometry, field decomposition, and data requirements

The practical UD-RM formulation is posed for an ocean-bottom acquisition with receivers on the seabed $\Gamma_R$, a source carpet $\Gamma_S$ in the water layer, and a subsurface volume of interest $V$ below the seabed. The recorded fields are acoustic pressure $p(\mathbf{x},t)$ and vertical particle velocity $v_z(\mathbf{x},t)$, with one-way decompositions

$$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$

where “$+$” denotes down-going toward increasing depth and “$-$” denotes up-going toward decreasing depth. In the homogeneous water layer, a simple characteristic-variable separation at a horizontal boundary uses the local water impedance $\Gamma_R$0,

$\Gamma_R$1

In practice, the separation may be carried out by PZ summation or more general one-way operators such as slowness-dependent filters (Wang et al., 2024).

The required data are correspondingly specific. UD-RM requires multi-component seabed receiver data at $\Gamma_R$2 and either dual-sensor sources at $\Gamma_R$3 or single-sensor sources for which source-deghosted signatures have been estimated by a model-based preprocessing step. The method utilizes only the down-going component of the receiver-side wavefield, which is then separated further into up-/down-going components at the source side. In the 2025 formulation, the same operational requirement is stated as up/down separation at both source and receiver sides in processing, together with amplitude-consistent preprocessing, direct-arrival traveltimes from a smooth migration model, and construction of the direct focusing function $\Gamma_R$4 (Wang et al., 29 Jul 2025).

The physical assumptions in the practical seabed formulation are acoustic, heterogeneous density and velocity below $\Gamma_R$5, a homogeneous water layer with density $\Gamma_R$6 and velocity $\Gamma_R$7, isotropy, and losslessness, with the free surface treated as a perfect pressure-release boundary with reflection coefficient $\Gamma_R$8. Free-surface multiples are not discarded; they are included explicitly in the theory, and exactness is claimed under these stated boundary conditions. The later imaging paper preserves the acoustic, heterogeneous, isotropic, lossless setting and likewise treats free-surface effects through Rayleigh boundary relationships and multi-dimensional deconvolution rather than by suppressing them a priori (Wang et al., 2024, Wang et al., 29 Jul 2025).

3. Coupled UD-RM equations and focusing functions

The transition from conventional RM to UD-RM is mediated by an “upside-down” reciprocity transformation. In the seabed formulation, one begins from a representation theorem in a domain bounded above by the source carpet $\Gamma_R$9, with vanishing lower boundary by the Sommerfeld radiation condition. After separating the direct arrival $\Gamma_R$0 and the free-surface components, the derivation yields

$\Gamma_R$1

Multiplication by $\Gamma_R$2 identifies the source-level reflection response

$\Gamma_R$3

and yields the practical relation

$\Gamma_R$4

which acts as the source-side counterpart of multi-dimensional deconvolution for free-surface multiples (Wang et al., 2024).

Using this reciprocity and the source-side data, the UD-RM system is written in the compact operator form

$\Gamma_R$5

where $\Gamma_R$6 denotes removal of the direct arrival, $\Gamma_R$7 is the total down-going focusing function, and $\Gamma_R$8 is the up-going focusing function. To isolate the causal parts, a traveltime-based window operator $\Gamma_R$9 is applied, yielding the reduced system for $\Gamma_S$0 and the coda $\Gamma_S$1 of the down-going focusing function: $\Gamma_S$2 The direct focusing term $\Gamma_S$3 is supplied by a smooth background model, and the reconstructed fields are obtained by substituting the retrieved focusing functions back into the unwindowed operator relation (Wang et al., 29 Jul 2025).

The imaging condition used in the later formulation is

$\Gamma_S$4

with $\Gamma_S$5 a small window around zero lag. This imaging step is explicitly tied to the claim that UD-RM produces images free from artifacts caused by both internal and surface-related multiples, because the redatumed up-/down-going fields have been reconstructed from focusing functions rather than from primary-only fields (Wang et al., 29 Jul 2025).

4. Relation to Rayleigh integrals and Marchenko representations without internal up/down decomposition

The 2021 work on Green’s function representations without up/down decomposition provides a broader theoretical setting for understanding why Rayleigh-type single-sided Marchenko schemes can be formulated without imposing an internal one-way decomposition throughout the medium. In the acoustic case, for a lower half-space bounded by a single horizontal acquisition boundary $\Gamma_S$6, the core full-field representation is

$\Gamma_S$7

where $\Gamma_S$8 is the reflection response at the boundary and $\Gamma_S$9 is a focusing function related to a source-free solution of the same wave operator. Receiver redatuming of the homogeneous Green’s function then gives the Rayleigh-type integral

$V$0

and combining source and receiver redatuming yields a double boundary integral over $V$1 (Wapenaar et al., 2021).

The significance of these representations for UD-RM is structural rather than terminological. They show that single-sided redatuming can be cast as a Rayleigh-type boundary integral weighted by full-field focusing functions rather than by half-space Green’s functions, and that the decomposition can be confined to the acquisition boundary rather than imposed inside the medium. In the elastodynamic case, the corresponding fields become matrices and tensors, with traction operators replacing scalar normal derivatives, but the same architecture persists: a focusing matrix $V$2 or $V$3, a reflection response matrix $V$4, and homogeneous Green’s functions recovered by single-sided integrals (Wapenaar et al., 2021).

This suggests a broader class of UD-RM formulations than the practical seabed acoustic scheme alone. In particular, the 2021 derivations retain refracted and interior evanescent components while only neglecting evanescent waves at the acquisition boundary. That does not remove the need for boundary decomposition in the seabed UD-RM workflow, but it indicates that “UD-RM” need not be conceptually restricted to the characteristic-variable acoustic setting; it can also be viewed as part of a family of single-sided Rayleigh-Marchenko redatuming schemes driven by full-field focusing functions.

5. Spectral and inverse-scattering formulation for Rayleigh waves

A distinct, more mathematical interpretation of UD-RM arises from the inverse problem for the time-harmonic Rayleigh system in an isotropic flat elastic half-space with a traction-free boundary. In that setting, the displacement satisfies the Navier equations

$V$5

and after tangential Fourier transform the Rayleigh system becomes a coupled depth-dependent ODE system for the tangential and normal components. The inverse-scattering analysis applies the Markushevich substitution, which converts the original boundary-value problem into a matrix Sturm–Liouville equation

$V$6

with a boundary condition containing the spectral parameter,

$V$7

The transformed potential $V$8 is non-symmetric, and the associated Jost function

$V$9

identifies the spectral data through the Weyl matrix

$p(\mathbf{x},t)$0

whose poles encode guided modes and whose jump across the branch cut gives the continuous spectral density (Hoop et al., 2021).

In the original inverse problem, the key reconstruction device is a Gel'fand–Levitan type integral equation for a kernel $p(\mathbf{x},t)$1,

$p(\mathbf{x},t)$2

from which the transformed potential $p(\mathbf{x},t)$3 is recovered. The main uniqueness theorem states that if the Jost functions coincide for all $p(\mathbf{x},t)$4 on the physical sheet and for any pair of distinct frequencies $p(\mathbf{x},t)$5, then the Lamé-type potential $p(\mathbf{x},t)$6 is uniquely recovered; with $p(\mathbf{x},t)$7 and $p(\mathbf{x},t)$8 known, one obtains $p(\mathbf{x},t)$9 and then recovers $v_z(\mathbf{x},t)$0 and $v_z(\mathbf{x},t)$1 from

$v_z(\mathbf{x},t)$2

Thus the spectral data at two frequencies determine the elastic parameters in the slab beneath the traction-free boundary (Hoop et al., 2021).

A synthesis connecting this inverse-scattering framework to a UD-RM methodology interprets the Weyl solution as a two-component focusing function and the Gel'fand–Levitan equation as the analog of a Marchenko kernel equation for Rayleigh waves. In that interpretation, “upgoing” and “downgoing” focusing components are defined via $v_z(\mathbf{x},t)$3, and the focusing function $v_z(\mathbf{x},t)$4 is recovered from

$v_z(\mathbf{x},t)$5

Because the original article is an inverse problem for the Rayleigh system rather than a seabed redatuming paper, this connection is interpretive rather than terminological. Even so, it suggests a frequency-domain elastic UD-RM in which surface spectral data, encoded in Jost or Weyl objects, drive a Gel'fand–Levitan reconstruction of focusing functions and, ultimately, of the Lamé parameters themselves (Hoop et al., 2021).

6. Algorithms, acceleration, validation, and limitations

In the practical seabed workflow, UD-RM proceeds through calibration, receiver-side up/down separation, source-side separation or source deghosting, subtraction of the direct arrival, assembly of the boundary operators on $v_z(\mathbf{x},t)$6, construction of the window operator from first-arrival traveltimes, solution of the windowed UD-RM system for $v_z(\mathbf{x},t)$7 and $v_z(\mathbf{x},t)$8, reconstruction of the redatumed Green’s functions, and imaging. For dense receiver layouts, least-squares solvers such as LSQR are used; for sparse or irregular receiver geometries, the problem becomes underdetermined and the literature introduces sparsity-promoting inversion in a sliding linear Radon domain, solved by FISTA. In the synthetic examples of the practical paper, LSQR converged in approximately $v_z(\mathbf{x},t)$9 iterations for dense receivers, whereas FISTA used approximately $$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$0 iterations for sparse receivers (Wang et al., 2024).

The synthetic validations establish the operational claims of the method. In the syncline model, UD-RM focusing functions obtained from dual-source data closely matched those from classical Marchenko without free surface; middle-trace comparisons of redatumed Green’s functions showed near-perfect alignment for dual-source data, and imaging produced clean, amplitude-consistent images and angle gathers. With sparse receivers at $$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$1 and $$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$2 of the original array, UD-RM still yielded satisfactory images, while FISTA-based inversion suppressed illumination artifacts more effectively than LSQR, particularly at $$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$3 receiver density. In the SEG/EAGE Overthrust model, UD-RM suppressed both free-surface and internal multiple artifacts and matched the quality of classical Marchenko imaging performed on a dataset without free surface and with co-located sources and receivers (Wang et al., 2024).

The principal computational bottleneck is the per-focal-point estimation of focusing functions. The 2025 work addresses this by training a U-Net on a small subset of focal points whose focusing functions were pre-computed by the conventional iterative scheme. The network input is the initial pair of subsurface wavefields

$$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$4

and the output is the predicted pair of focusing functions. The loss is a windowed normalized MSE,

$$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$5

and the architecture is a five-level U-Net with channels $$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$6, Leaky ReLU with negative slope $$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$7, batch normalization, dropout $$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$8, and positional embeddings derived from normalized focal-point coordinates (Wang et al., 29 Jul 2025).

The reported computational gains are substantial. For the synthetic dense case with $$p(\mathbf{x},\omega)=p^+(\mathbf{x},\omega)+p^-(\mathbf{x},\omega), \qquad v_z(\mathbf{x},\omega)=v_z^+(\mathbf{x},\omega)+v_z^-(\mathbf{x},\omega),$$9 focal points, conventional LSQR required approximately $+$0 per point and approximately $+$1 total, whereas the U-Net required approximately $+$2 for a $+$3 train-plus-validation split, approximately $+$4 for $+$5, and approximately $+$6 for $+$7, corresponding to speed-ups of $+$8, $+$9, and $-$0. For the synthetic sparse case, conventional FISTA required approximately $-$1 per point and approximately $-$2 total, while the U-Net reduced this to approximately $-$3, $-$4, and $-$5 for the same training fractions, corresponding to $-$6, $-$7, and $-$8 speed-ups. In the Volve field experiment with $-$9 focal points, LSQR required approximately $\Gamma_R$00 total, whereas the U-Net required approximately $\Gamma_R$01, $\Gamma_R$02, and $\Gamma_R$03, corresponding to $\Gamma_R$04, $\Gamma_R$05, and $\Gamma_R$06 speed-ups (Wang et al., 29 Jul 2025).

These validations also delimit the method’s current scope. UD-RM requires multi-component receivers and either dual-sensor sources or accurate source deghosting; deghosting errors contaminate the separated source-side fields. It depends on reasonably accurate first-arrival traveltimes for the window operator, and strong anisotropy or attenuation fall outside the stated theory. In the inverse-scattering Rayleigh-wave formulation, further limitations include flat half-space geometry, isotropic elasticity, depth-dependent Lamé parameters, density fixed to $\Gamma_R$07, a homogeneous half-space beneath the slab, strong ellipticity, smoothness assumptions, and spectral restrictions such as simple poles and the absence of poles with $\Gamma_R$08 except possibly at $\Gamma_R$09. Extensions to solid–fluid interfaces, anisotropy, layered $\Gamma_R$10D media, curved topography, multi-mode coupling, and density variation are discussed only as possible generalizations rather than as completed UD-RM formulations (Hoop et al., 2021, Wang et al., 2024).

Taken together, the cited works place UD-RM at the intersection of practical seabed redatuming, single-sided Rayleigh-Marchenko theory, and matrix inverse scattering. In its operational form it is a source-carpet-integrated, full-wavefield redatuming scheme for ocean-bottom data; in its broader theoretical setting it can be viewed as part of a family of single-sided focusing methods that use Rayleigh-type boundary integrals and, in the Rayleigh-wave case, admit a spectral Gel'fand–Levitan interpretation.